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1. The stable exotic Cuntz algebras are higher-rank graph algebras

   https://doi.org/10.1090/bproc/180  

   Boersema, Jeffrey ; Browne, Sarah ; Gillaspy, Elizabeth ( March 2024 , Proceedings of the American Mathematical Society, Series B)

   For each odd integer$n \geq 3$, we construct a rank-3 graph$\Lambda _n$with involution$\gamma _n$whose real$C^*$-algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n)$is stably isomorphic to the exotic Cuntz algebra$\mathcal E_n$. This construction is optimal, as we prove that a rank-2 graph with involution$(\Lambda ,\gamma )$can never satisfy$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )\sim _{ME} \mathcal E_n$, and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math.10(2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution$(\Lambda , \gamma )$whose real$C^*$-algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )$is stably isomorphic to the suspension$S \mathbb {R}$. In the Appendix, we show that the$i$-fold suspension$S^i \mathbb {R}$is stably isomorphic to a graph algebra iff$-2 \leq i \leq 1$.

    

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2. 𝐿₁-distortion of Wasserstein metrics: A tale of two dimensions

   https://doi.org/10.1090/btran/143  

   Baudier, F. ; Gartland, C. ; Schlumprecht, Th. ( August 2023 , Transactions of the American Mathematical Society, Series B)

   By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid$\{0,1,\dots , n\}^2$has$L_1$-distortion bounded below by a constant multiple of$\sqrt {\log n}$. We provide a new “dimensionality” interpretation of Kislyakov’s argument, showing that if$\{G_n\}_{n=1}^\infty$is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number$\delta \in [2,\infty )$, then the 1-Wasserstein metric over$G_n$has$L_1$-distortion bounded below by a constant multiple of$(\log |G_n|)^{\frac {1}{\delta }}$. We proceed to compute these dimensions for$\oslash$-powers of certain graphs. In particular, we get that the sequence of diamond graphs$\{\mathsf {D}_n\}_{n=1}^\infty$has isoperimetric dimension and Lipschitz-spectral dimension equal to 2, obtaining as a corollary that the 1-Wasserstein metric over$\mathsf {D}_n$has$L_1$-distortion bounded below by a constant multiple of$\sqrt {\log | \mathsf {D}_n|}$. This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of$L_1$-embeddable graphs whose sequence of 1-Wasserstein metrics is not$L_1$-embeddable.

    

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3. A modular construction of unramified 𝑝-extensions of ℚ(ℕ^{1/𝕡})

   https://doi.org/10.1090/bproc/141  

   Lang, Jaclyn ; Wake, Preston ( October 2022 , Proceedings of the American Mathematical Society, Series B)

   We show that for primes$N, p \geq 5$with$N \equiv -1 \bmod p$, the class number of$\mathbb {Q}(N^{1/p})$is divisible by$p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when$N \equiv -1 \bmod p$, there is always a cusp form of weight$2$and level$\Gamma _0(N^2)$whose$\ell$th Fourier coefficient is congruent to$\ell + 1$modulo a prime above$p$, for all primes$\ell$. We use the Galois representation of such a cusp form to explicitly construct an unramified degree-$p$extension of$\mathbb {Q}(N^{1/p})$.

    

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4. A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures

   https://doi.org/10.1090/tran/8976  

   Livshyts, Galyna ( June 2023 , Transactions of the American Mathematical Society)

   We show that for any even log-concave probability measure$\mu$on$\mathbb {R}^n$, any pair of symmetric convex sets$K$and$L$, and any$\lambda \in [0,1]$,$\begin{equation*} \mu ((1-\lambda ) K+\lambda L)^{c_n}\geq (1-\lambda ) \mu (K)^{c_n}+\lambda \mu (L)^{c_n}, \end{equation*}$where$c_n\geq n^{-4-o(1)}$. This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Richard J. Gardner and Artem Zvavitch [Tran. Amer. Math. Soc. 362 (2010), pp. 5333–5353]; Andrea Colesanti, Galyna V. Livshyts, Arnaud Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139]). Moreover, our bound improves for various special classes of log-concave measures.

    

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5. Residual intersections and linear powers

   https://doi.org/10.1090/btran/127  

   Eisenbud, David ; Huneke, Craig ; Ulrich, Bernd ( October 2023 , Transactions of the American Mathematical Society, Series B)

   If$I$is an ideal in a Gorenstein ring$S$, and$S/I$is Cohen-Macaulay, then the same is true for any linked ideal$I’$; but such statements hold for residual intersections of higher codimension only under restrictive hypotheses, not satisfied even by ideals as simple as the ideal$L_{n}$of minors of a generic$2 \times n$matrix when$n>3$.

   In this paper we initiate the study of a different sort of Cohen-Macaulay property that holds for certain general residual intersections of the maximal (interesting) codimension, one less than the analytic spread of$I$. For example, suppose that$K$is the residual intersection of$L_{n}$by$2n-4$general quadratic forms in$L_{n}$. In this situation we analyze$S/K$and show that$I^{n-3}(S/K)$is a self-dual maximal Cohen-Macaulay$S/K$-module with linear free resolution over$S$.

   The technical heart of the paper is a result about ideals of analytic spread 1 whose high powers are linearly presented.

    

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